K5 and K3,3 are Toroidal Penny Graphs

Abstract

In this article we emphasize on the connection between two fields of study that are Penny Graphs, and the Optimal Packing of Spheres on the Flat Torus. We give a brief litterature overview on related results in the fields of planar graphs, penny graphs, toroidal penny graphs and spherical codes.We also show that $K5$ and $K_{3,3}$ are penny graphs on the flat square torus.

Figures (6)

• Figure 1: left: the complete graph $K_5$, right: the complete bipartite graph $K_{3,3}$
• Figure 2: https://commons.wikimedia.org/w/index.php?curid=56426404
• Figure 3: left: $K_5$ penny graph embedding on the unit flat square torus, right: $K_5$ penny graph embedding on a 3x3 toroidal tiling
• Figure 4: left: Planar embedding of the octahedral graph, right: Penny graph embedding of the octahedral graph
• Figure 5: left: $K_{3,3}$ penny graph embedding on the unit flat square torus, right: $K_{3,3}$ penny graph embedding on a 3x3 toroidal tiling

Theorems & Definitions (1)

• Conjecture 1